Editing Group ring

{{Short description|The set of finitely supported functions from a group to a ring}}
{{about|the algebraic group ring of a group|the case of a topological group|group algebra of a topological group}}

In [[algebra]], a '''group ring''' is a [[free module]] and at the same time a [[Ring (mathematics)|ring]], constructed in a natural way from any given ring and any given [[Group (mathematics)|group]]. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring.

If the ring is commutative then the group ring is also referred to as a '''group algebra''', for it is indeed an [[Algebra over a ring|algebra]] over the given ring. A group algebra over a field has a further structure of a [[Hopf algebra]]; in this case, it is thus called a [[group Hopf algebra]].

The apparatus of group rings is especially useful in the theory of [[group representation]]s.

==Definition==
Let <math>G</math> be a group, written multiplicatively, and let <math>R</math> be a ring. The group ring of <math>G</math> over <math>R</math>, which we will denote by <math>R[G]</math>, or simply <math>RG</math>, is the set of mappings <math>f\colon G \to R</math> of [[Support (mathematics)#Generalization|finite support]] (<math>f(g)</math> is nonzero for only finitely many elements <math>g</math>), where the module scalar product <math>\alpha f </math> of a scalar <math>\alpha</math> in <math>R</math> and a mapping <math>f</math> is defined as the mapping <math>x \mapsto \alpha \cdot f(x)</math>, and the module group sum of two mappings <math>f</math> and <math>g</math> is defined as the mapping <math>x \mapsto f(x) + g(x)</math>. To turn the additive group <math>R[G]</math> into a ring, we define the product of <math>f</math> and <math>g</math> to be the mapping
:<math>x\mapsto\sum_{uv=x}f(u)g(v)=\sum_{u\in G}f(u)g(u^{-1}x).</math>
The summation is legitimate because <math>f</math> and <math>g</math> are of finite support, and the ring axioms are readily verified.

Some variations in the notation and terminology are in use. In particular, the mappings such as <math>f : G \to R</math> are sometimes<ref>Milies & Sehgal (2002), pp. 129 and 131.</ref> written as what are called "formal linear combinations of elements of <math>G</math> with coefficients in <math>R</math>
":
:<math>\sum_{g\in G}f(g) g,</math>
or simply
:<math>\sum_{g\in G}f_g g.</math><ref name=Milies>Milies & Sehgal (2002), p. 131.</ref>

Note that if the ring <math>R</math> is in fact a field <math>K</math>, then the module structure of the group ring <math>RG</math> is in fact a vector space over <math>K</math>.

==Examples==
1. Let {{nowrap|1=''G'' = ''C''<sub>3</sub>}}, the [[cyclic group]] of order 3, with generator <math>a</math> and identity element 1<sub>''G''</sub>. An element ''r'' of '''C'''[''G''] can be written as

:<math>r = z_0 1_G + z_1 a + z_2 a^2\,</math>

where ''z''<sub>0</sub>, ''z''<sub>1</sub> and ''z''<sub>2</sub> are in '''C''', the [[complex numbers]]. This is the same thing as a [[polynomial ring]] in variable <math>a</math> such that <math>a^3=a^0=1</math> i.e. '''C'''[''G''] is isomorphic to the ring '''C'''[<math>a</math>]/<math>(a^3-1)</math>.

Writing a different element ''s'' as <math>s=w_0 1_G +w_1 a +w_2 a^2</math>, their sum is

:<math>r + s = (z_0+w_0) 1_G + (z_1+w_1) a + (z_2+w_2) a^2\,</math>

and their product is

:<math>rs = (z_0w_0 + z_1w_2 + z_2w_1) 1_G  +(z_0w_1 + z_1w_0 + z_2w_2)a +(z_0w_2 + z_2w_0 + z_1w_1)a^2.</math>

Notice that the identity element 1<sub>''G''</sub> of ''G'' induces a canonical embedding of the coefficient ring (in this case '''C''') into '''C'''[''G'']; however strictly speaking the multiplicative identity element of '''C'''[''G''] is 1⋅1<sub>''G''</sub> where the first ''1'' comes from '''C''' and the second from ''G''. The additive identity element is zero.

When ''G'' is a non-commutative group, one must be careful to preserve the order of the group elements (and not accidentally commute them) when multiplying the terms.

2. The ring of [[Laurent polynomial]]s over a ring ''R'' is the group ring of the [[infinite cyclic group]] '''Z''' over ''R''.

3. Let ''Q'' be the [[quaternion group]] with elements <math>\{e, \bar{e}, i, \bar{i}, j, \bar{j}, k, \bar{k}\}</math>. Consider the group ring '''R'''''Q'', where '''R''' is the set of real numbers. An arbitrary element of this group ring is of the form

:<math>x_1 \cdot e + x_2 \cdot \bar{e} + x_3 \cdot i + x_4 \cdot \bar{i} + x_5 \cdot j + x_6 \cdot \bar{j} + x_7 \cdot k + x_8 \cdot \bar{k}</math>
where <math>x_i </math> is a real number.

Multiplication, as in any other group ring, is defined based on the group operation. For example,

:<math>\begin{align} \big(3 \cdot e + \sqrt{2} \cdot i \big)\left(\frac{1}{2} \cdot \bar{j}\right) &= (3 \cdot e)\left(\frac{1}{2} \cdot \bar{j}\right) + (\sqrt{2} \cdot i)\left(\frac{1}{2} \cdot \bar{j}\right)\\
&= \frac{3}{2} \cdot \big((e)(\bar{j})\big) + \frac{\sqrt{2}}{2} \cdot \big((i)(\bar{j})\big)\\
&= \frac{3}{2} \cdot \bar{j} + \frac{\sqrt{2}}{2} \cdot \bar{k}
\end{align}.</math>

Note that '''R'''''Q'' is not the same as the skew field of [[quaternions]] over '''R'''. This is because the skew field of quaternions satisfies additional relations in the ring, such as <math>-1 \cdot i = -i</math>, whereas in the group ring '''R'''''Q'', <math>-1\cdot i</math> is not equal to <math>1\cdot \bar{i}</math>. To be more specific, the group ring '''R'''''Q'' has dimension 8 as a real [[vector space]], while the skew field of quaternions has dimension 4 as a [[real vector space]].

4. Another example of a non-abelian group ring is <math>\mathbb{Z}[\mathbb{S}_3]</math> where <math>\mathbb{S}_3</math> is the symmetric group on 3 letters. This is not an integral domain since we have <math>[1 - (12)]*[1+(12)] = 1 -(12)+(12) -(12)(12) = 1 - 1 = 0</math> where the element <math>(12)\in \mathbb{S}_3</math> is the [[Cyclic_permutation#Transpositions|transposition]] that swaps 1 and 2. Therefore the group ring need not be an integral domain even when the underlying ring is an integral domain.

==Some basic properties==
Using 1 to denote the multiplicative identity of the ring ''R'', and denoting the group unit by 1<sub>''G''</sub>, the ring ''R''[''G''] contains a subring isomorphic to ''R'', and its group of invertible elements contains a subgroup isomorphic to ''G''. For considering the [[indicator function]] of {1<sub>''G''</sub>}, which is the vector ''f'' defined by
:<math>f(g)= 1\cdot 1_G + \sum_{g\not= 1_G}0 \cdot g= \mathbf{1}_{\{1_G\}}(g)=\begin{cases}
1 & g = 1_G \\
0 & g \ne 1_G
\end{cases},</math>
the set of all scalar multiples of ''f'' is a subring of ''R''[''G''] isomorphic to ''R''. And if we map each element ''s'' of ''G'' to the indicator function of {''s''}, which is the vector ''f'' defined by
:<math>f(g)= 1\cdot s + \sum_{g\not= s}0 \cdot g= \mathbf{1}_{\{s\}}(g)=\begin{cases}
1 & g = s \\
0 & g \ne s
\end{cases}</math>
the resulting mapping is an injective group homomorphism (with respect to multiplication, not addition, in ''R''[''G'']).

If ''R'' and ''G'' are both commutative (i.e., ''R'' is commutative and ''G'' is an [[abelian group]]), ''R''[''G''] is commutative.

If ''H'' is a [[subgroup]] of ''G'', then ''R''[''H''] is a [[subring]] of ''R''[''G''].  Similarly, if ''S'' is a subring of ''R'', ''S''[''G''] is a subring of ''R''[''G''].

If ''G'' is a finite group of order greater than 1, then ''R''[''G''] always has [[zero divisors]]. For example, consider an element ''g'' of ''G'' of order {{math|1= {{!}}''g''{{!}} = ''m'' > 1}}. Then 1 − ''g'' is a zero divisor:
<math display="block">
(1 - g)(1 + g+\cdots+g^{m-1}) = 1 - g^m = 1 - 1 =0.
</math>

For example, consider the group ring '''Z'''[''S''<sub>3</sub>] and the element of order 3 ''g'' = (123). In this case,
<math display="block">
(1 - (123))(1 + (123)+ (132)) = 1 - (123)^3 = 1 - 1 =0.
</math>
A related result: If the group ring <math> K[G] </math> is [[Prime ring|prime]], then ''G'' has no nonidentity finite normal subgroup (in particular, ''G'' must be infinite).

Proof: Considering the [[contrapositive]], suppose <math> H </math> is a nonidentity finite normal subgroup of <math> G </math>. Take <math> a = \sum_{h \in H} h </math>. Since <math> hH = H </math> for any <math> h \in H </math>, we know <math> ha = a </math>, therefore <math> a^2 = \sum_{h \in H} h a = |H|a </math>. Taking <math> b = |H|\,1 - a </math>, we have <math> ab = 0 </math>. By normality of <math> H </math>, <math> a </math> commutes with a basis of <math> K[G] </math>, and therefore
:<math> aK[G]b=K[G]ab=0 </math>.
And we see that <math> a,b </math> are not zero, which shows <math> K[G] </math> is not prime. This shows the original statement.

==Group algebra over a finite group==
Group algebras occur naturally in the theory of [[group representation]]s of [[finite group]]s. The group algebra ''K''[''G''] over a field ''K'' is essentially the group ring, with the field ''K'' taking the place of the ring. As a set and vector space, it is the [[free vector space]] on ''G'' over the field ''K''. That is, for ''x'' in ''K''[''G''],
:<math>x=\sum_{g\in G} a_g g.</math>

The [[algebra over a field|algebra]] structure on the vector space is defined using the multiplication in the group:
:<math>g \cdot h = gh,</math>
where on the left, ''g'' and ''h'' indicate elements of the group algebra, while the multiplication on the right is the group operation (denoted by juxtaposition).

Because the above multiplication can be confusing, one can also write the [[basis vector]]s of ''K''[''G''] as ''e''<sub>''g''</sub> (instead of ''g''), in which case the multiplication is written as:
:<math>e_g \cdot e_h = e_{gh}.</math>

===Interpretation as functions===
Thinking of the [[free vector space]] as ''K''-valued functions on ''G'', the algebra multiplication is [[convolution]] of functions.

While the group algebra of a ''finite'' group can be identified with the space of functions on the group, for an infinite group these are different. The group algebra, consisting of ''finite'' sums, corresponds to functions on the group that vanish for [[cofinitely]] many points; topologically (using the [[discrete topology]]), these correspond to functions with [[compact support]].

However, the group algebra ''K''[''G''] and the space of functions {{nowrap|1=''K''<sup>''G''</sup> := Hom(''G'', ''K'')}} are dual: given an element of the group algebra

:<math>x = \sum_{g\in G} a_g g</math>

and a function on the group {{nowrap|''f'' : ''G'' → ''K''}} these pair to give an element of ''K'' via

:<math>(x,f) = \sum_{g\in G} a_g f(g),</math>

which is a well-defined sum because it is finite.

=== Representations of a group algebra ===
Taking ''K''[''G''] to be an abstract algebra, one may ask for [[group representation|representations]] of the algebra acting on a ''K-''vector space ''V'' of dimension ''d''. Such a representation

:<math>\tilde{\rho}:K[G]\rightarrow \mbox{End} (V)</math>

is an algebra homomorphism from the group algebra to the algebra of [[endomorphism]]s of ''V'', which is isomorphic to the ring of ''d × d'' matrices: <math>\mathrm{End}(V)\cong M_{d}(K) </math>. Equivalently, this is a [[module (mathematics)|left ''K''[''G'']-module]] over the abelian group ''V''.

Correspondingly, a group representation

:<math>\rho:G\rightarrow \mbox{Aut}(V),</math>

is a group homomorphism from ''G'' to the group of linear automorphisms of ''V'', which is isomorphic to the [[general linear group]] of invertible matrices: <math>\mathrm{Aut}(V)\cong \mathrm{GL}_d(K) </math>. Any such representation induces an algebra representation

:<math>\tilde{\rho}:K[G]\rightarrow \mbox{End}(V),</math>

simply by letting <math>\tilde{\rho}(e_g) = \rho(g)</math> and extending linearly. Thus, representations of the group correspond exactly to representations of the algebra, and the two theories are essentially equivalent.

=== Regular representation ===
{{Main|Regular representation}}
The group algebra is an algebra over itself; under the correspondence of representations over ''R'' and ''R''[''G''] modules, it is the [[regular representation]] of the group.

Written as a representation, it is the representation ''g'' {{mapsto}} ''ρ''<sub>''g''</sub> with the action given by <math>\rho(g)\cdot e_h = e_{gh}</math>, or

:<math>\rho(g)\cdot r =  \sum_{h\in G} k_h \rho(g)\cdot e_h = \sum_{h\in G} k_h e_{gh}. </math>

===Semisimple decomposition===
The dimension of the vector space ''K''[''G''] is just equal to the number of elements in the group. The field ''K'' is commonly taken to be the complex numbers '''C''' or the reals '''R''', so that one discusses the group algebras '''C'''[''G''] or '''R'''[''G''].

The group algebra '''C'''[''G''] of a finite group over the complex numbers is a [[semisimple ring]]. This result, [[Maschke's theorem]], allows us to understand '''C'''[''G''] as a finite [[Product of rings|product]] of [[matrix ring]]s with entries in '''C'''. Indeed, if we list the complex [[irreducible representation]]s of ''G'' as ''V<sub>k</sub>'' for ''k'' = 1, . . . , ''m'', these correspond to [[group homomorphism]]s <math>\rho_k: G\to \mathrm{Aut}(V_k)</math> and hence to algebra homomorphisms <math>\tilde\rho_k: \mathbb{C}[G]\to \mathrm{End}(V_k)</math>. Assembling these mappings gives an algebra isomorphism
:<math>\tilde\rho : \mathbb{C}[G] \to \bigoplus_{k=1}^m \mathrm{End}(V_k)
\cong \bigoplus_{k=1}^m M_{d_k}(\mathbb{C}),  </math>
where ''d<sub>k</sub>'' is the dimension of ''V<sub>k</sub>''. The subalgebra of '''C'''[''G''] corresponding to End(''V<sub>k</sub>'') is the [[Ideal (ring theory)|two-sided ideal]] generated by the [[Idempotent (ring theory)|idempotent]]
:<math>\epsilon_k = \frac{d_k}{|G|}\sum_{g\in G}\chi_k(g^{-1})\,g,   </math>
where <math>\chi_k(g)=\mathrm{tr}\,\rho_k(g)  </math> is the [[Character theory|character]] of ''V<sub>k</sub>''. These form a complete system of orthogonal idempotents, so that <math>\epsilon_k^2 =\epsilon_k  </math>, <math>\epsilon_j \epsilon_k = 0  </math> for ''j ≠ k'', and <math>1 = \epsilon_1+\cdots+\epsilon_m   </math>. The isomorphism <math>\tilde\rho</math> is closely related to [[Fourier transform on finite groups]].

For a more general field ''K,'' whenever the [[characteristic (algebra)|characteristic]] of ''K'' does not divide the order of the group ''G'', then ''K''[''G''] is semisimple. When ''G'' is a finite [[abelian group]], the group ring ''K''[G] is commutative, and its structure is easy to express in terms of [[root of unity|roots of unity]].

When ''K'' is a field of characteristic ''p'' which divides the order of ''G'', the group ring is ''not'' semisimple: it has a non-zero [[Jacobson radical]], and this gives the corresponding subject of [[modular representation theory]] its own, deeper character.

===Center of a group algebra===
The [[center of a group|center]] of the group algebra is the set of elements that commute with all elements of the group algebra:
:<math>\mathrm{Z}(K[G]) := \left\{ z \in K[G] : \forall r \in K[G], zr = rz \right\}.</math>

The center is equal to the set of [[class function]]s, that is the set of elements that are constant on each conjugacy class
:<math>\mathrm{Z}(K[G]) = \left\{ \sum_{g \in G} a_g g :  \forall g,h \in G, a_g = a_{h^{-1}gh}\right\}.</math>

If {{nowrap|1=''K'' = '''C'''}}, the set of irreducible [[character theory|characters]] of ''G'' forms an orthonormal basis of Z(''K''[''G'']) with respect to the inner product
:<math>\left \langle \sum_{g \in G} a_g g, \sum_{g \in G} b_g g \right \rangle = \frac{1}{|G|} \sum_{g \in G} \bar{a}_g b_g.</math>

==Group rings over an infinite group==
Much less is known in the case where ''G'' is countably infinite, or uncountable, and this is an area of active research.<ref>{{cite journal|author=Passman, Donald S.|author-link=Donald S. Passman|title=What is a group ring?|journal=Amer. Math. Monthly|volume=83|year=1976|issue=3 |pages=173–185|url=http://www.maa.org/programs/maa-awards/writing-awards/what-is-a-group-ring|doi=10.2307/2977018|jstor=2977018 }}</ref> The case where ''R'' is the field of complex numbers is probably the one best studied. In this case, [[Irving Kaplansky]] proved that if ''a'' and ''b'' are elements of '''C'''[''G''] with {{nowrap|1=''ab'' = 1}}, then {{nowrap|1=''ba'' = 1}}. Whether this is true if ''R'' is a field of positive characteristic remains unknown.

A long-standing [[Kaplansky's conjectures|conjecture of Kaplansky]] (~1940) says that if ''G'' is a [[torsion-free group]], and ''K'' is a field, then the group ring ''K''[''G''] has no non-trivial [[zero divisor]]s. This conjecture is equivalent to ''K''[''G''] having no non-trivial [[nilpotent]]s under the same hypotheses for ''K'' and ''G''.

In fact, the condition that ''K'' is a field can be relaxed to any ring that can be embedded into an [[integral domain]].

The conjecture remains open in full generality, however some special cases of torsion-free groups have been shown to satisfy the zero divisor conjecture. These include:

* Unique product groups (e.g. [[orderable group]]s, in particular [[free group]]s)
* [[Elementary amenable group]]s (e.g. [[virtually abelian group]]s)
* Diffuse groups – in particular, groups that act freely isometrically on ''R''-trees, and the fundamental groups of surface groups except for the fundamental groups of direct sums of one, two or three copies of the projective plane.

The case where ''G'' is a [[topological group]] is discussed in greater detail in the article [[Group algebra of a locally compact group]].

==Category theory==

===Adjoint===
[[Category theory|Categorically]], the group ring construction is [[left adjoint]] to "[[group of units]]"; the following functors are an [[adjoint functors|adjoint pair]]:
:<math>R[-]\colon \mathbf{Grp} \to R\mathbf{\text{-}Alg}</math>
:<math>(-)^\times\colon R\mathbf{\text{-}Alg} \to \mathbf{Grp}</math>
where <math>R[-]</math> takes a group to its group ring over ''R'', and <math>(-)^\times</math> takes an ''R''-algebra to its group of units.

When {{nowrap|1=''R'' = '''Z'''}}, this gives an adjunction between the [[category of groups]] and the [[category of rings]], and the unit of the adjunction takes a group ''G'' to a group that contains trivial units: {{nowrap|1=''G'' × {±1} = {±''g''}.}} In general, group rings contain nontrivial units. If ''G'' contains elements ''a'' and ''b'' such that <math>a^n=1</math> and ''b'' does not normalize <math>\langle a\rangle</math> then the square of

:<math>x=(a-1)b \left (1+a+a^2+...+a^{n-1} \right )</math>

is zero, hence <math>(1+x)(1-x)=1</math>. The element {{nowrap|1 + ''x''}} is a unit of infinite order.

=== Universal property ===
The above adjunction expresses a universal property of group rings.<ref name=Milies/><ref>{{Cite web|url=https://ncatlab.org/nlab/show/group+algebra#general|title=group algebra in nLab|website=ncatlab.org|access-date=2017-11-01}}</ref> Let {{var|R}} be a  (commutative) ring, let {{var|G}} be a group, and let {{var|S}} be an {{var|R}}-algebra. For any group homomorphism <math>f:G\to S^\times</math>, there exists a unique {{var|R}}-algebra homomorphism <math>\overline{f}:R[G]\to S</math> such that <math>\overline{f}^\times \circ i=f</math> where {{var|i}} is the inclusion

:<math>\begin{align}
  i:G &\longrightarrow R[G] \\
  g &\longmapsto 1_Rg
\end{align}</math>

In other words, <math>\overline{f}</math> is the unique homomorphism making the following diagram commute:

:[[Image:Group ring UMP.svg|200px]]

Any other ring satisfying this property is [[List of mathematical jargon|canonically]] isomorphic to the group ring.

=== Hopf algebra ===
The group algebra ''K''[''G''] has a natural structure of a [[Hopf algebra]]. The comultiplication is defined by <math>\Delta(g)=g\otimes g </math>, extended linearly, and the antipode is <math>S(g)=g^{-1}</math>, again extended linearly.

===Generalizations===
The group algebra generalizes to the [[monoid ring]] and thence to the [[category algebra]], of which another example is the [[incidence algebra]].

==Filtration==
{{Expand section|date=December 2008}}
If a group has a [[length function]] – for example, if there is a choice of generators and one takes the [[word metric]], as in [[Coxeter group]]s – then the group ring becomes a [[filtered algebra]].

==See also==
* [[Group algebra of a locally compact group]]
* [[Monoid ring]]
* [[Kaplansky's conjectures]]

===Representation theory===
* [[Group representation]]
* [[Regular representation]]

===Category theory===
* [[Categorical algebra]]
* [[Group of units]]
* [[Incidence algebra]]
*[[Quiver (mathematics)|Quiver algebra]]

== Notes ==
{{Reflist}}

==References==
* {{springer|id=G/g045220|title=Group algebra|author=A. A. Bovdi}}
* Milies, César Polcino; Sehgal, Sudarshan K. ''[https://books.google.com/books?id=7m9P9hM4pCQC An introduction to group rings]''. Algebras and applications, Volume 1. Springer, 2002. {{ISBN|978-1-4020-0238-0}}
* [[Charles W. Curtis]], [[Irving Reiner]]. [https://books.google.com/books?id=RKwjeZKMr8oC ''Representation theory of finite groups and associative algebras''], Interscience  (1962)
* D.S. Passman, [https://books.google.com/books?id=2xrSHX-rpGMC ''The algebraic structure of group rings''], Wiley  (1977)

{{DEFAULTSORT:Group Ring}}
[[Category:Ring theory]]
[[Category:Representation theory of groups]]
[[Category:Harmonic analysis]]

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